Microwave Engineering Europe - December 2008 - (Page 16) 16 X-PARAMETERS X-Parameters: The new paradigm for measurement, modeling, and design of nonlinear RF and microwave components By David E. Root, Jason Horn, Loren Betts, Chad Gillease, Jan Verspecht* Agilent Technologies, Inc., Santa Rosa, CA (*Jan Verspecht, bvba, Opwijk, Belgium) or more than 40 years, S-parameters, or scattering parameters, have been among the most important of all the foundations of microwave theory and techniques. S-parameters are easy to measure at high frequencies with a vector network analyzer (VNA). A well-calibrated S-parameter measurement represents intrinsic properties of the DUT, independent of the VNA system used to characterize it. These DUT properties (gain, loss, reflection coefficient, etc.) are familiar, intuitive, and important. S-parameters are still commonly used for nonlinear devices such as transistors and amplifiers. The problem, often forgotten or taken for granted, is that S-parameters only describe properly the behavior of a nonlinear component in response to small signal stimuli for which the device can be approximated as a linear component at a fixed DC, or static, operating point. The need for a rigorous, yet practical solution for characterization, modeling, and design of nonlinear components at high frequencies has never been more urgent. The communications revolution is inexorably forcing active components into more and more strongly nonlinear regimes of operation. X-parameters are the rigorous supersets of S-parameters that are applicable to linear and nonlinear components, excited by small and large-signal conditions. They reduce exactly to S-parameters in the small-signal limit. Unlike S-parameters, however, X-parameters contain detailed and useful information including magnitudes and phases of distortion products generated by the nonlinear component in response to large-signal conditions. X-parameters enable a predictable and hierarchical design methodology for cascades of nonlinear components from knowledge of their constituent X-parameters. X-parameters can be applied wherever linear S-parameters are currently used for active or nonlinear devices and components but with more benefit. F Figure 1: PHD Framework: The model is formulated in the frequency domain, and maps incident waves (A) to scattered waves (B). The complete knowledge of magnitude and phase of incident and scattered waves at all harmonics is exactly equivalent to complete knowledge of the time domain waveforms, so the full nonlinear input-output characteristics of the device are captured. PHD framework: Origins of X-parameters The large-signal steady-state behavior of a RF or microwave component is completely determined by its nonlinear constitutive relations, B(A), which specify the output, or scattered, B-waves as complex-valued nonlinear functions of the complex-valued incident A-waves. This is shown for the case of a multi-stage amplifier in Figure 1 and written in Equation 1. The first index denotes the port number (e = 1, 2). The second index represents the order of the harmonic response (f = 1, 2,…, N). For simplicity, Figure 1 illustrates the case where the spectral components of incident signals define a harmonic grid. Note here the input and output phasors have components at each harmonic of the fundamental signal, in contrast to the single small-amplitude complex phasor that is used for the definition of S-parameters. It is important to consider signals incident at different ports simultaneously, since superposition does not apply. This means the formalism accounts for interactions (mixing) among the various frequency components of the incident signals by the nonlinear device. Equation 1: Bef = Fef (DC, A11, A12, , A21, A22, ) www.mwee.com When KCL and KVL are expressed in A-B space, they yield two equivalent conditions relating the incident and scattered waves of two or more components at the node at which they are joined. Cascadablility follows directly from these conditions. PHD simulation components cascade with any other conventional linear (e.g. S-parameters) and nonlinear models in the simulator, including other PHD components. Identification of the nonlinear constitutive relations requires magnitude and crossfrequency phase information of multiple spectral components simultaneously, measurements beyond those possible with a conventional VNA. The Agilent Nonlinear Vector Network Analyzer (NVNA) is a new instrument capable of measuring these quantities with the high dynamic range of a state-of-the-art network analyzer [1]. It is the only commercial instrument for which X-parameter measurement and extraction is implemented. X-parameter concept: spectral linearization The DUT B(A) constitutive relations are simplified to make their identification practical. The most drastic simplification leads to the familiar S-parameter case. S-parameters result Microwave Engineering Europe ● December 2008 ● http://www.mwee.com
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